Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806551 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,15,16,13);(9,11)(10,13)(12,16)(14,15)(17,18,22,27)(19,25,30,32)(20,24,28,26)(21,31,29,23);(2,5)(6,8)(10,14)(11,16)(17,19,28,21,22,30,20,29)(18,23,24,32,27,31,26,25);(10,14)(13,15)(17,20,22,28)(18,24,27,26)(19,29,30,21)(23,32,31,25);(1,3,4,7)(2,6,5,8)(17,21,28,30,22,29,20,19)(18,25,24,23,27,32,26,31)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fhi,d^2=i,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6739824);